A081090 a(n)^2 + 1 = A081089(n), where A081089(n) = A081088(n+1)/A081088(n); involves the partial quotients of a series of continued fractions that sum to unity.

0, 2, 5, 52, 13525, 9512132552, 1223751003414213892335125, 14245051228051734585272181044575005954679284643762013257552

Offset

1,2

Comments

log(a(n+1))/log(a(n)) -> 1+sqrt(2). The 8th term has 59 digits, while the 9th term has 141 digits.

sum(n>0, a(n)/a(n+1) ) = 1/2; the ratio of the terms a(n+1)/a(n), for n>1, form the convergents of the continued fraction series described by A081088; thus a(n+1) = A081088(n)*a(n) + a(n-1), for n>1. - Paul D. Hanna, Mar 05 2003

Links

Table of n, a(n) for n=1..8.

Formula

a(n) = a(n-2)*(a(n-1)^2 + 1) for n>3, with a(1)=0, a(2)=2, a(3)=5. Also, a(n)*a(n-1) = A081088(n) for n>2, a(n) = a(n-2)*A081089(n-1) for n>2.

Crossrefs

Cf. A081086, A081088, A081089.

Sequence in context: A216079 A206584 A268286 * A330282 A071880 A071882

Adjacent sequences: A081087 A081088 A081089 * A081091 A081092 A081093

Keyword

nonn,easy

Author

Hans Havermann and Paul D. Hanna, Mar 05 2003

Status

approved